Asymptotic expansions for the error probabilities of repeated significance tests about a normal mean are developed. Use of the expansions appears to result in substantially improved numerical accuracy, when compared to the use of the leading term, at least in some important special cases. The expansions are sufficiently refined to show the effect of some simple modifications of the basic procedure, such as requiring an initial sample size.
"Asymptotic Expansions for the Error Probabilities of Some Repeated Significance Tests." Ann. Statist. 10 (3) 895 - 908, September, 1982. https://doi.org/10.1214/aos/1176345879